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AdvancesinAppliedMathematics
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,2022,11(7),4089-4109
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117437
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ExistenceofSign-ChangingSolutionsfor
aFractionalChoquardEquation
JinhuaGao
DepartmentofMathematics,YunnanNormalUniversity,KunmingYunnan
Received:Jun.1
st
,2021;accepted:Jun.24
th
,2022;published:Jul.1
st
,2022
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DOI:10.12677/aam.2022.117437
p
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Abstract
Withanimportantphysicalbackground,thefractionalChoquardequationhasat-
tractedgreatattentionfromthefieldofnonlinearanalysisinrecentyears.Inthis
paper,westudythefollowingfractionalChoquardequation
(
−
∆)
s
u
+
V
(
x
)
u
= (
|
x
|
−
µ
∗|
u
|
p
)
|
u
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p
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u,
in
R
N
,
(P)
where
s
∈
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,
N
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3
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µ
∈
(0
,N
)
,
2
<p<
2
N
−
µ
N
−
2
s
,“
∗
”standsfortheconvolutionand
(
−
∆)
s
isthefractionalLaplacianoperator.BycombiningtheEkelandvariationalprinciple
withtheimplicitfunctiontheorem,we provethattheproblem
(
P
)
possessesoneleast
energysign-changingsolution
w
.Moreover,weshowthattheenergyof
w
isstrictly
largerthanthegroundstateenergyandlessthantwicethegroundstateenergy.
Keywords
FractionalLaplacian,Sign-ChangingSolutions,ChoquardEquation
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.1174374092
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1
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k
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DOI:10.12677/aam.2022.1174374093
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f
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DOI:10.12677/aam.2022.1174374094
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DOI:10.12677/aam.2022.1174374095
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DOI:10.12677/aam.2022.1174374096
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.
y
²
:
c
∈
R
¿
…
{
w
j
}
j
∈
N
´
H
¥
˜
‡
S
,
¦
j
→∞
ž
,
J
(
w
j
)
→
c,J
0
(
w
j
)
→
0
.
·
‚
k
c
+1+
k
w
j
k
=
J
(
w
j
)
−
1
2
p
h
J
0
(
w
j
)
,w
j
i
= (
1
2
−
1
2
p
)
Z
R
N
+1
+
y
1
−
2
s
|∇
w
j
|
2
dxdy
+
Z
R
N
V
(
x
)
|
w
j
(
x,
0)
|
2
dx
≥
1
4
k
w
j
k
2
.
(2.2)
Ï
d
,
S
{
w
j
}
j
∈
N
3
H
¥
k
.
.
Š
â
(2.1),
3
f
¿Â
e
,
·
‚
Ø
”
b
3
H
þ
,w
j
w
0
,
3
L
p
(
R
N
)
þ
,w
j
(
x,
0)
→
w
0
(
x,
0)
,
w
j
(
x,
0)
→
w
0
(
x,
0) a.e.
x
∈
R
N
.
Ø
d
ƒ
,
Š
â
[28],
•
3
g
(
x,
0)
∈
L
p
(
R
N
)
¦
|
w
j
(
x,
0)
|≤
g
(
x,
0)a.e.
x
∈
R
N
,
∀
j
∈
N
.
Š
â
V
‚
›
›
Â
ñ
½
n
,
·
‚
Z
R
N
(
|
x
|
−
µ
∗|
w
j
(
x,
0)
|
p
)
|
w
j
(
x,
0)
|
p
dx
→
Z
R
N
(
|
x
|
−
µ
∗|
w
0
(
x,
0)
|
p
)
|
w
0
(
x,
0)
|
p
dx,
(2.3)
…
j
→∞
ž
,
Z
R
N
(
|
x
|
−
µ
∗|
w
j
(
x,
0)
|
p
)
|
w
j
(
x,
0)
|
p
−
2
w
j
(
x,
0)
w
0
(
x,
0)
dx
→
Z
R
N
(
|
x
|
−
µ
∗|
w
0
(
x,
0)
|
p
)
|
w
0
(
x,
0)
|
p
dx.
(2.4)
DOI:10.12677/aam.2022.1174374097
A^
ê
Æ
?
Ð
p
7
u
Ï
d
,
Š
â
h
J
0
(
w
j
)
,w
j
i→
0
Ú
(2.3),
j
→∞
ž
,
Z
R
N
+1
+
y
1
−
2
s
|∇
w
j
|
2
dxdy
+
Z
R
N
(
V
(
x
)
|
w
j
(
x,
0)
|
2
)
dx
→
Z
R
N
(
|
x
|
−
µ
∗|
w
0
(
x,
0)
|
p
)
|
w
0
(
x,
0)
|
p
dx.
(2.5)
d
,
Š
â
{
w
j
}
Ú
(2.4)
f
Â
ñ
5
,
j
→∞
ž
,
Z
R
N
+1
+
y
1
−
2
s
|∇
w
0
|
2
+
Z
R
N
V
(
x
)
|
w
0
(
x,
0)
|
2
dx
=
Z
R
N
(
|
x
|
−
µ
∗|
w
0
(
x,
0)
|
p
)
|
w
0
(
x,
0)
|
p
dx,
(2.6)
Ï
d
,
j
→∞
ž
,
k
w
j
k→k
w
0
k
,
(2.7)
ù
L
²
3
H
¥
{
w
j
}
j
∈
N
r
Â
ñ
,
ù
Ò
¤
Ú
n
4
y
²
.
w
,
,
ù
‡
•
¼
J
ä
k
ì
´
(
,
=
•
3
ρ,r>
0
¦
inf
J
(
∂B
r
)
≥
ρ
…
é
u
?
¿
w
6
=0,
lim
t
→∞
J
(
tw
) =
−∞
.
Ï
d
,
·
‚
k
Ú
n
5.
µ
∈
(0
,N
)
,
2
<p<
2
N
−
µ
N
−
2
s
.
K
•
§
(1.1)
k
˜
‡
Ä
)
.
·
‚
y
3
y
²
Ä
)
ØC
Ò
(
ë
„
[8],[9]).
Ú
n
6.
µ
∈
(0
,N
)
,
2
<p<
2
N
−
µ
N
−
2
s
.
K
•
§
(1.1)
?
Û
Ä
)
u
∈
H
s
(
R
N
)
ØC
Ò
.
y
²
:
e
u
=
w
(
x,
0)
´
(1.1)
Ä
)
,
K
w
∈
X
s
(
R
N
+1
)
´
¯
K
(1.7)
)
.
l
,
|
w
|
•
´
˜
‡
)
.
Ï
d
,
|
u
|
•
´
˜
‡
Ä
)
.
§
÷
v
•
§
(
−
∆)
s
|
u
|
+
V
(
x
)
|
u
|
= (
|
x
|
−
µ
∗|
u
|
p
)
|
u
|
p
−
1
, x
∈
R
N
.
¦
^
u
−
Š
•
(1.1)
¥
u
¼
ê
,
·
‚
0 =
Z
R
N
(
−
∆)
s
2
u
(
−
∆)
s
2
u
−
dx
+
Z
R
N
V
(
x
)
uu
−
dx
+
Z
R
N
(
|
x
|
−
µ
∗|
u
|
p
)
|
u
|
p
−
2
uu
−
dx
=
Z
R
N
|
(
−
∆)
s
2
u
−
|
2
dx
+
Z
R
N
V
(
x
)
|
u
−
|
2
dx
+
Z
R
N
(
|
x
|
−
µ
∗|
u
|
p
)
|
u
−
|
p
dx
≥
Z
R
N
|
(
−
∆)
s
2
u
−
|
2
dx
+
Z
R
N
V
(
x
)
|
u
−
|
2
dx
≥
0
.
Ï
d
,
u
−
= 0.
l
,
u
≥
0.
d
,
e
é
u
x
0
∈
R
N
,
u
(
x
0
) = 0,
K
(
−
∆)
s
u
(
x
0
) = 0.
2
Š
â
(1.5),
(
−
∆)
s
u
(
x
0
) =
−
C
N,s
2
Z
R
N
u
(
x
0
+
y
)+
u
(
x
0
−
y
)
−
2
u
(
x
0
)
|
y
|
N
+2
s
dy,
Ï
d
Z
R
N
u
(
x
0
+
y
)+
u
(
x
0
−
y
)
|
y
|
N
+2
s
dy
= 0
,
DOI:10.12677/aam.2022.1174374098
A^
ê
Æ
?
Ð
p
7
u
du
u
´
š
K
,
Œ
u
≡
0.
ù
†
u
6
= 0
ƒ
g
ñ
.
u
´
.
Ï
d
,
u
ØC
Ò
.
e
¡
Ú
n
L
²
8
Ü
M
š
˜
.
Ú
n
7.
µ
∈
(0
,N
)
,
2
<p<
2
N
−
µ
N
−
2
s
.
K
é
u
?
¿
÷
v
w
±
6
=0
w
∈
H
,
•
3
•
˜˜
é
(
d,t
) = (
d
w
,t
w
)
∈
R
+
×
R
+
,
¦
d
w
w
+
+
t
w
w
−
∈M
,
=
M6
=
∅
.
y
²
:
·
‚
æ
^
[7]
Ú
[20]
¥
g
Ž
,
Ä
k
,
·
‚
½
Â
¼
ê
G
(
d
1
,d
2
) :=
J
(
d
1
p
1
w
+
+
d
1
p
2
w
−
)
.
d
,
é
u
0
<s
0
,β<N
Ú
0
<s
0
+
β<N
,
l
[20,
í
Ø
5.10]
Ñ
(
|
x
|
s
0
−
N
∗|
x
|
β
−
N
)(
y
) :=
Z
R
N
|
z
|
s
0
−
N
|
y
−
z
|
β
−
N
dz
=
C
N
−
s
0
−
β
C
s
0
C
β
C
s
0
+
β
C
N
−
s
0
C
N
−
β
|
y
|
s
0
+
β
−
N
,
(2.8)
Ù
¥
C
s
0
:=
π
−
s
0
2
Γ(
s
0
2
)
.
s
0
=
β
=
N
−
µ
2
ž
,
•
•
B
,
·
‚
P
C
(
N,µ
) :=
C
s
0
+
β
C
N
−
s
0
C
N
−
β
C
N
−
s
0
−
β
C
s
0
C
β
,
Š
â
(2.8),
|
x
|
−
µ
=
C
(
N,µ
)
|
x
|
−
N
+
µ
2
∗|
x
|
−
N
+
µ
2
.
Ï
d
Z
R
N
|
x
|
−
µ
∗
(
|
d
1
p
1
w
+
(
x,
0)+
d
1
p
2
w
−
(
x,
0)
|
p
)
|
d
1
p
1
w
+
(
x,
0)+
d
1
p
2
w
−
(
x,
0)
|
p
dx
=
C
(
N,µ
)
Z
R
N
|
x
|
−
N
+
µ
2
∗
|
x
|
−
N
+
µ
2
∗|
d
1
p
1
w
+
(
x,
0)+
d
1
p
2
w
−
(
x,
0)
|
p
!
|
d
1
p
1
w
+
(
x,
0)
+
d
1
p
2
w
−
(
x,
0)
|
p
dx
=
C
(
N,µ
)
Z
R
N
Z
R
N
|
x
−
z
|
−
N
+
µ
2
|
z
|
−
N
+
µ
2
∗|
d
1
p
1
w
+
(
z,
0)+
d
1
p
2
w
−
(
z,
0)
|
p
dz
|
d
1
p
1
w
+
(
x,
0)
+
d
1
p
2
w
−
(
x,
0)
|
p
dx
=
C
(
N,µ
)
Z
R
N
|
z
|
−
N
+
µ
2
∗|
d
1
p
1
w
+
(
x,
0)+
d
1
p
2
w
−
(
x,
0)
|
p
Z
R
N
|
x
−
z
|
−
N
+
µ
2
|
d
1
p
1
w
+
(
z,
0)
+
d
1
p
2
w
−
(
z,
0)
|
p
dxdz
=
C
(
N,µ
)
Z
R
N
|
z
|
−
N
+
µ
2
∗|
d
1
p
1
w
+
(
z,
0)+
d
1
p
2
w
−
(
z,
0)
|
p
2
dz
=
C
(
N,µ
)
Z
R
N
|
z
|
−
N
+
µ
2
∗
(
d
1
|
w
+
(
z,
0)
|
p
+
d
2
|
w
−
(
z,
0)
|
p
)
2
dz.
(2.9)
DOI:10.12677/aam.2022.1174374099
A^
ê
Æ
?
Ð
p
7
u
Š
â
(2.9),
·
‚
k
G
(
d
1
,d
2
) =
d
2
p
1
2
Z
R
N
+1
+
y
1
−
2
s
|∇
w
+
|
2
dxdy
+
Z
R
V
(
x
)
|
w
+
(
x,
0)
|
2
dx
+
d
2
p
2
2
Z
R
N
+1
+
y
1
−
2
s
|∇
w
−
|
2
dxdy
+
Z
R
N
V
(
x
)
|
w
−
(
x,
0)
|
2
dx
−
1
2
p
Z
R
N
|
x
|
−
µ
∗
(
|
d
1
p
1
w
+
(
x,
0)+
d
1
p
2
w
−
(
x,
0)
|
p
)
|
d
1
p
1
w
+
(
x,
0)+
d
1
p
2
w
−
(
x,
0)
|
p
dx
=
d
2
p
1
2
k
w
+
k
2
+
d
2
p
2
2
k
w
−
k
2
−
C
N,µ
2
p
Z
R
N
|
x
|
−
N
+
µ
2
∗
d
1
|
w
+
(
x,
0)
|
p
+
d
2
|
w
−
(
x,
0)
|
p
!
2
dx.
du
G
(
d
1
,d
2
)
´
ë
Y
¼
ê
,
¿
…
G
(
d
1
,d
2
) =
d
2
p
1
2
k
w
+
k
2
+
d
2
p
2
2
k
w
−
k
2
−
d
2
1
2
p
Z
R
N
(
|
x
|
−
µ
∗|
w
+
(
x,
0)
|
p
)
|
w
+
(
x,
0)
|
p
dx
−
d
1
d
2
p
Z
R
N
(
|
x
|
−
µ
∗|
w
+
(
x,
0)
|
p
)
|
w
−
(
x,
0)
|
p
dx
−
d
2
2
2
p
Z
R
N
(
|
x
|
−
µ
∗|
w
−
(
x,
0)
|
p
)
|
w
−
(
x,
0)
|
p
dx
≤
1
2
(
d
2
1
+
d
2
2
)
1
p
(
k
w
+
k
2
+
k
w
−
k
2
)
−
1
2
p
min
{
A
1
,A
2
}
(
d
2
1
+
d
2
2
)
,
Ù
¥
A
1
=
R
R
N
(
|
x
|
−
µ
∗|
w
+
(
x,
0)
|
p
)
|
w
+
(
x,
0)
|
p
dx,
A
2
=
R
R
N
(
|
x
|
−
µ
∗|
w
−
(
x,
0)
|
p
)
|
w
−
(
x,
0)
|
p
dx.
|
(
d
1
,d
2
)
|→
+
∞
ž
,
G
(
d
1
,d
2
)
→−∞
,
G
k
˜
‡
Û
•
Œ
Š
:
(
a,b
)
∈
R
+
×
R
+
.
ù
p
·
‚
^
G
î
‚
]
5
.
¯¢
þ
,
P
F
(
d
1
,d
2
) =
d
2
p
1
2
k
w
+
k
2
+
d
2
p
2
2
k
w
−
k
2
,
T
(
d
1
,d
2
) =
Z
R
N
|
x
|
−
N
+
µ
2
∗
d
1
|
w
+
(
x,
0)
|
p
+
d
2
|
w
−
(
x,
0)
|
p
!
2
dx,
K
(
F
00
d
1
d
2
)
2
−
F
00
d
1
d
1
F
00
d
2
d
2
<
0
.
DOI:10.12677/aam.2022.1174374100
A^
ê
Æ
?
Ð
p
7
u
Ï
d
,
F
´
î
‚
]
¼
ê
.
d
,
-
(
d
1
,d
2
),(
t
1
,t
2
)
∈
R
+
×
R
+
,
λ
∈
(0
,
1),
K
T
(
λ
(
d
1
,d
2
)+(1
−
λ
)(
t
1
,t
2
)) =
T
(
λd
1
+(1
−
λ
)
t
1
,λd
2
+(1
−
λ
)
t
2
)
=
Z
R
N
|
x
|
−
N
+
µ
2
∗
(
λd
1
+(1
−
λ
)
t
1
|
w
+
(
x,
0)
|
p
+(
λd
2
+(1
−
λ
)
t
2
)
|
w
−
(
x,
0)
|
p
!
2
dx
=
Z
R
N
λ
|
x
|
−
N
+
µ
2
∗
d
1
|
w
+
(
x,
0)
|
p
+
d
2
|
w
−
(
x,
0)
|
p
+(1
−
λ
)
|
x
|
−
N
+
µ
2
∗
t
1
|
w
+
(
x,
0)
|
p
+
t
2
|
w
−
(
x,
0)
|
p
!
2
dx
<λ
Z
R
N
|
x
|
−
N
+
µ
2
∗
d
1
|
w
+
(
x,
0)
|
p
+
d
2
|
w
−
(
x,
0)
|
p
!
2
dx
+(1
−
λ
)
Z
R
N
|
x
|
−
N
+
µ
2
∗
t
1
|
w
+
(
x,
0)
|
p
+
t
2
|
w
−
(
x,
0)
|
p
!
2
dx
=
λT
(
d
1
,d
2
)+(1
−
λ
)
T
(
t
1
,t
2
)
,
ù
`
²
G
´
î
‚
]
¼
ê
,
d
d
Ñ
(
Ø
(
a,b
)
∈
R
+
×
R
+
´
•
˜
Û
4
Œ
Š
:
,
¿
…
∇
G
(
a,b
) =
(0
,
0),
ù
`
²
M6
=
∅
.
d
(2.2),
e
(
Ø
¤
á
.
Ú
n
8.
J
(
w
)
3
M
þ
e
•
k
.
…r
›
.
d
Ú
n
7,
·
‚
•
Ä
X
e
å
4
¯
K
m
:= inf
{
J
(
u
) :
w
∈M}
.
(2.10)
Ú
n
9.
m<
2
c
.
y
²
:
Š
â
Ú
n
5,
e
´
¯
K
(1.1)
˜
‡
Ä
)
,
=
J
0
(
e
) = 0
,J
(
e
) =
c,e>
0
.
η
⊂
C
∞
0
(
R
N
+1
,
[0
,
1])
•
˜
ä
¼
ê
,
¦
supp
η
∈
B
1
(0) :=
{
z
= (
x,y
)
∈
R
N
+1
:
|
z
|≤
1
}
,
3
B
1
2
(0)
þ
η
≡
1,
3
R
N
+1
\
B
1
(0)
þ
η
≡
0,
¿
…
|∇
η
|
<
1.
½
Â
w
R
(
x,y
) :=
η
(
x
R
,y
)
e
(
x,
0)
≥
0
,v
R
(
x,y
) :=
−
η
(
x
−
x
n
R
,y
)
e
(
x,
0)
≤
0
,
DOI:10.12677/aam.2022.1174374101
A^
ê
Æ
?
Ð
p
7
u
Ù
¥
R>
0,
x
n
= (0
,
0
,
···
,
0
,
3
R
).
Ï
d
,
·
‚
Œ
±
y
²
supp
w
R
(
x,y
)
∩
supp
v
R
(
x,y
) =
∅
,
ù
L
²
w
R
(
x,y
) +
v
R
(
x,y
)
6
= 0,
¿
…
w
R
(
x,y
)
6
=0,
v
R
(
x,y
)
6
= 0.
Ï
d
,
Š
â
Ú
n
7,
•
3
•
˜˜
é
(
d
w
R
,t
v
R
)
∈
R
+
×
R
+
¦
¯
u
R
:=
d
w
R
w
R
+
t
v
R
v
R
∈M
…
=
d
2
w
R
k
w
R
k
2
−
d
2
p
w
R
A
3
−
d
p
w
R
t
p
v
R
B
= 0
,
t
2
v
R
k
v
R
k
2
−
t
2
p
v
R
A
4
−
t
p
v
R,N
d
p
w
R
B
= 0
.
(2.11)
•
{
ü
å
„
,
·
‚
¦
^
±
e
Î
Ò
:
A
3
=
R
R
N
(
|
x
|
−
µ
∗|
w
R
|
p
)
|
w
R
|
p
dx,
B
=
R
R
N
(
|
x
|
−
µ
∗|
v
R
|
p
)
|
w
R
|
p
dx,
A
4
=
R
R
N
(
|
x
|
−
µ
∗|
v
R
|
p
)
|
v
R
|
p
dx.
(2.12)
d
,
Š
â
w
R
Ú
v
R
½
Â
,
y
1
−
2
s
|∇
(
w
R
−
e
)
|
2
+
V
(
x
)
|
w
R
−
e
|
2
≤
C
|∇
η
(
x
R
,y
)
|
2
e
2
+
|
η
(
x
R
,y
)
−
1
|
2
|∇
e
|
2
∈
L
1
(
R
N
)
.
a
q
/
y
1
−
2
s
|∇
(
v
R
+
e
)
|
2
+
V
(
x
)
|
v
R
(
x,
0)+
e
(
x,
0)
|
2
∈
L
1
(
R
N
)
.
Ï
d
,
Š
â
V
‚
›
›
Â
ñ
½
n
,
3
H
¥
,
R
→∞
ž
,
w
R
→
e,v
R
→−
e.
(2.13)
Ï
d
Z
R
N
(
|
x
|
−
µ
∗|
w
R
|
p
)
|
w
R
|
p
dx
→
Z
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx,
(2.14)
¿
…
Z
R
N
(
|
x
|
−
µ
∗|
v
R
|
p
)
|
v
R
|
p
dx
→
Z
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx.
(2.15)
e
5
,
·
‚
ò
y
²
R
→∞
ž
•
3
(
d
0
,t
0
)
∈
R
+
×
R
+
¦
d
w
R
→
d
0
,t
v
R
→
t
0
,
(
d
0
,t
0
)
∈
(0
,
1)
×
(0
,
1)
.
(2.16)
¯¢
þ
,
-
lim
R
→
+
∞
d
w
R
= +
∞
.
Š
â
(2.11)
Ú
(2.14)
·
‚
Ñ
0
≤
t
p
v
R
d
p
w
R
B
=
1
d
2
p
−
2
w
R
k
w
R
k
2
−
A
3
=
−
Z
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx
+
o
(1)
<
0
,
DOI:10.12677/aam.2022.1174374102
A^
ê
Æ
?
Ð
p
7
u
g
ñ
.
Ï
d
,
d
w
R
´
˜
—
k
.
.
a
q
u
(2.11)
Ú
(2.15)
y
²
,
t
v
R
•
´
˜
—
k
.
.
Ø
”
˜
„
5
,
·
‚
Œ
±
b
•
3
d
0
,t
0
∈
[0
,
∞
),
¦
R
→∞
ž
,
d
w
R
→
d
0
,t
v
R
→
t
0
.
e
d
0
= 0
½
t
0
= 0,
Š
â
(2.11)-(2.15),
Z
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx
2
+
o
(1) =
1
d
2
p
−
2
w
R
k
w
R
k
2
−
A
3
1
t
2
p
−
2
v
R
k
v
R
k
2
−
A
4
+
o
(1) = +
∞
,
ù
´
Ø
Œ
U
.
Ï
d
,
·
‚
(
d
0
,t
0
)
∈
R
+
×
R
+
.
Š
â
(2.13)
Ú
(2.14),
·
‚
Œ
±
d
2
w
R
k
e
k
2
−
d
2
p
w
R
R
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx
−
d
p
w
R
t
p
v
R
R
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx
= 0
,
t
2
v
R
k
e
k
2
−
t
2
p
v
R
R
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx
−
t
p
v
R
d
p
w
R
R
R
N
(
|
x
|
−
µ
∗|
e
|
p
)
|
e
|
p
dx
= 0
.
(
Ü
J
0
(
e
) = 0,
·
‚
1
d
2
p
−
2
w
R
−
1 =
t
p
v
R
d
p
w
R
,
1
t
2
p
−
2
v
R
−
1 =
d
p
w
R
t
p
v
R
.
Ï
d
,(
d
0
,t
0
)
∈
(0
,
1)
×
(0
,
1),(2.16)
y
.
Ï
d
,
·
‚
d
¯
u
R
∈M
Ú
(2.16)
Œ
,
R
→
+
∞
ž
,
m
≤
J
(¯
u
R
) = (
1
2
−
1
2
p
)
Z
R
N
+1
+
y
1
−
2
s
|∇
¯
u
R
|
2
dxdy
+
Z
R
N
V
(
x
)
|
¯
u
R
(
x,
0)
|
2
dx
= (
1
2
−
1
2
p
)(
d
2
w
R
k
w
R
k
2
+
t
2
v
R
k
v
R
k
2
)
= (
t
2
0
+
d
2
0
)(
1
2
−
1
2
p
)
k
e
k
2
+
o
(1)
= (
t
2
0
+
d
2
0
)
J
(
e
)+
o
(1)
= (
t
2
0
+
d
2
0
)
c
+
o
(1)
≤
2
c.
Ú
n
y
.
3.
C
Ò
)
!
Ì
‡
8
´
y
²
·
‚
Ì
‡
(
J
.
½
n
1
y
²
y
²
:
·
‚
Ä
k
y
²
å
4
¯
K
(2.10)
4
w
(
´
(1.1)
˜
‡
)
,
=
J
0
(
w
) = 0.
DOI:10.12677/aam.2022.1174374103
A^
ê
Æ
?
Ð
p
7
u
d
Ú
n
8
Ú
Ekeland
C
©
n
,
·
‚
Ø
”
b
•
3
4
z
S
{
w
n
}⊂M
,
¦
J
(
w
n
)
≤
m
+
1
n
,
J
(
v
)
≥
J
(
w
n
)
−
1
n
k
w
n
−
v
k
.
(3.1)
·
‚
Œ
±
y
²
S
{
w
±
n
}
Ñ
3
H
¥
˜
—
k
.
.
Ï
d
,
3
f
¿Â
e
,
3
H
¥
,w
±
n
w
±
,
3
L
t
(
R
N
)
¥
,
é
u
t
∈
[2
,
2
∗
s
)
,w
±
n
(
x,
0)
→
w
±
(
x,
0)
,
w
±
n
(
x,
0)
→
w
±
(
x,
0) a.e.
x
∈
R
N
.
d
,
d
Sobolev
;
i
\
(2.1),
·
‚
w
+
(
x,
0)
≥
0
,w
−
(
x,
0)
≤
0
,w
+
(
x,
0)
·
w
−
(
x,
0) = 0 a.e.
x
∈
R
N
.
Ï
d
,
•
y
²
ù
‡
½
n
,
·
‚
•
I
y
²
J
0
(
w
n
)
→
0.
e
5
,
é
u
?
¿
φ
∈
C
∞
0
(
R
N
)
Ú
z
‡
n
,
·
‚
½
Â
T
1
n
,T
2
n
∈
C
1
(
R
3
,
R
)
X
e
:
T
1
n
(
σ,k,l
) =
k
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
+
k
2
−
Z
R
N
(
|
x
|
−
µ
∗|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
+
|
p
)
|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
+
|
p
dx
−
Z
R
N
(
|
x
|
−
µ
∗|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
−
|
p
)
|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
+
|
p
dx,
T
2
n
(
σ,k,l
) =
k
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
−
k
2
−
Z
R
N
(
|
x
|
−
µ
∗|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
−
|
p
)
|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
−
|
p
dx
−
Z
R
N
(
|
x
|
−
µ
∗|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
+
|
p
)
|
(
w
n
+
σφ
+
kw
+
n
+
lw
−
n
)
−
|
p
dx,
K
T
1
n
(0
,
0
,
0) =
T
2
n
(0
,
0
,
0) = 0.
d
,
Š
â
c
¡
Î
Ò
,
e
^
w
n
O
†
A
1
,A
2
¥
w
,
^
w
−
n
Ú
w
+
n
©
O
O
†
B
¥
v
R
Ú
w
R
,
K
∂T
1
n
(
σ,k,l
)
∂k
|
(0
,
0
,
0)
= 2(1
−
p
)
A
1
+(2
−
p
)
B,
∂T
2
n
(
σ,k,l
)
∂l
|
(0
,
0
,
0)
= 2(1
−
p
)
A
1
+(2
−
p
)
B
¿
…
∂T
1
n
(
σ,k,l
)
∂l
|
(0
,
0
,
0)
=
∂T
2
n
(
σ,k,l
)
∂k
|
(0
,
0
,
0)
=
−
pB.
d
,
Š
â
[12],[20,
½
n
9.8],
B
2
<A
1
A
2
≤
A
1
+
A
2
2
,
(3.2)
DOI:10.12677/aam.2022.1174374104
A^
ê
Æ
?
Ð
p
7
u
-
J
(0
,
0
,
0) =
∂T
1
n
(
σ,k,l
)
∂k
∂T
1
n
(
σ,k,l
)
∂l
∂T
2
n
(
σ,k,l
)
∂k
∂T
2
n
(
σ,k,l
)
∂l
.
d
(3.2),
k
det
J
(0
,
0
,
0) = 4(1
−
p
)
2
A
1
A
2
+2(1
−
p
)(2
−
P
)(
A
1
+
A
2
)
B
+4(1
−
p
)
B
2
≥
8(1
−
p
)(2
−
p
)
B
2
>
0
.
Š
â
Û
¼
ê
½
n
,
•
3
˜
‡
S
{
σ
n
}⊂
R
+
Ú
k
n
(
σ
)
,l
n
(
σ
)
∈
C
1
(
−
σ
n
,σ
n
)
÷
v
k
n
(0) = 0
,l
n
(0) = 0,
…
T
1
n
(
σ,k
n
(
σ
)
,l
n
(
σ
)) = 0
, T
2
n
(
σ,k
n
(
σ
)
,l
n
(
σ
)) = 0
.
Ï
d
,
é
u
∀
σ
∈
(
−
σ
n
,σ
n
),
φ
n,σ
:=
w
n
+
σφ
+
k
n
(
σ
)
w
+
n
+
l
n
(
σ
)
w
−
n
∈M
.
d
,
Š
â
(3.1),
J
(
φ
n,σ
)
−
J
(
w
n
)
≥−
1
n
k
σφ
+
k
n
(
σ
)
w
+
n
+
l
n
(
σ
)
w
−
n
k
.
(3.3)
d
(3.3),
(
Ü
÷
v
h
J
0
(
w
n
)
,w
±
n
i
= 0
V
Ð
m
ª
,
σ
h
J
0
(
w
n
)
,φ
i
+
o
(
k
σφ
+
k
n
(
σ
)
w
+
n
+
l
n
(
σ
)
w
−
n
k
)
≥−
1
n
k
σφ
+
k
n
(
σ
)
w
+
n
+
l
n
(
σ
)
w
−
n
k
σ
.
(3.4)
du
{
w
n
}
3
H
¥
´
˜
—
k
.
…
det
J
(0
,
0
,
0)
>
0,
·
‚
{
k
0
n
(0)
}
Ú
{
l
0
n
(0)
}
Ñ
´
˜
—
k
.
.
Ï
d
σ
→
0
ž
,
o
(
k
σφ
+
k
n
(
σ
)
w
+
n
+
l
n
(
σ
)
w
−
n
k
)
σ
→
0
.
Ï
d
,
Š
â
(3.4),
σ
→
0
ž
,
|h
J
0
(
w
n
)
,σ
i|≤
C
n
,
Ù
¥
C
´
˜
‡
†
n
Ã
'
~
ê
.
Ï
d
,
·
‚
J
0
(
w
n
)
→
0,
ù
`
²
J
0
(
w
)=0.
d
,
é
u
w
n
∈M
,
Š
â
(2.7),
·
‚
k
w
±
n
k→k
w
±
k
.
Ï
d
,
3
H
¥
,
n
→∞
ž
,
w
±
n
→
w
±
.
d
,
Š
â
Hardy-Littlewood-Sobolev
Ø
ª
Ú
h
J
0
(
w
n
)
,w
±
n
i
= 0,
k
w
+
n
k
2
=
Z
R
N
(
|
x
|
−
µ
∗|
w
+
n
(
x,
0)
|
p
)
|
w
+
n
(
x,
0)
|
p
dx
+
Z
R
N
(
|
x
|
−
µ
∗|
w
−
n
(
x,
0)
|
p
)
|
w
+
n
(
x,
0)
|
p
dx
≤
C
1
k
w
+
n
k
2
p
+
C
2
k
w
−
n
k
p
k
w
+
n
k
p
.
(3.5)
DOI:10.12677/aam.2022.1174374105
A^
ê
Æ
?
Ð
p
7
u
a
q
/
,
k
k
w
−
n
k
2
=
Z
R
N
(
|
x
|
−
µ
∗|
w
−
n
(
x,
0)
|
p
)
|
w
−
n
(
x,
0)
|
p
dx
+
Z
R
N
(
|
x
|
−
µ
∗|
w
+
n
(
x,
0)
|
p
)
|
w
−
n
(
x,
0)
|
p
dx
≤
C
1
k
w
−
n
k
2
p
+
C
2
k
w
+
n
k
p
k
w
−
n
k
p
.
(3.6)
Ï
d
,
•
3
˜
‡
~
ê
>
0,
¦
é
u
¤
k
n
∈
N
,
k
w
±
n
k≥
.
¤
±
k
w
±
k≥
>
0,
ù
`
²
w
±
6
= 0
¿
…
h
J
0
(
w
)
,w
±
i
= 0.
Ï
d
,
w
∈M
,
J
(
w
)=
m
,
=
w
´
J
˜
‡
.
:
.
Ï
d
,
w
´
¯
K
(1.1)
˜
‡
C
Ò
)
.
y
.
.
d
½
n
1,
·
‚
•
¯
K
(1.1)
k
˜
‡
4
U
þ
C
Ò
)
w
.
·
‚
y
3
y
²
w
U
þ
î
‚
Œ
u
Ä
U
þ
,
î
‚
u
Ä
U
þ
ü
.
½
n
2
y
²
y
²
:
Š
â
½
n
1
y
²
,(1.1)
•
3
4
U
þ
C
Ò
)
w
.
e
5
,
·
‚
ä
ó
J
(
w
) =
m
= inf
{
J
(
v
)
|
v
±
6
= 0
,J
0
(
v
) = 0
}
.
(3.7)
¯¢
þ
,
du
J
(
w
) =
m
,
w
´
J
.
:
¿
…
w
±
6
= 0,
·
‚
J
(
w
) =
m
≥
inf
{
J
(
v
)
|
v
±
6
= 0
,J
0
(
v
) = 0
}
.
d
,
Š
â
{
J
(
v
)
|
v
±
6
= 0
,J
0
(
v
) = 0
}⊂M6
=
∅
,
Œ
inf
{
J
(
v
)
|
v
±
6
= 0
,J
0
(
v
) = 0
}≥
inf
w
∈M
J
(
w
) =
m,
ù
Ò
`
²
(3.7)
¤
á
.
Ï
d
,
d
M⊂N
,
·
‚
k
J
(
w
)=
m
≥
c
.
Š
â
Ú
n
6,
¯
K
(1.1)
z
‡
Ä
)
ØC
Ò
,
d
d
Ñ
(
Ø
J
(
w
) =
m>c
.
Ï
d
,
Š
â
Ú
n
9,
·
‚
k
c<m<
2
c
.
y
.
.
©
Ì
‡
|
^
C
©•{
ï
Ä
˜
a
©
ê
Choquard
•
§
C
Ò
)
•
3
5
.
·
‚
ï
Ä
˜
a
‘
k
š
Û
Ü
š
‚
5
‘
©
ê
Choquard
•
§
,
T
•
§
A
:
´
Ó
ž
Ñ
y
©
ê
Ž
f
š
Û
Ü
5
Ú
š
‚
5
‘
š
Û
Ü
5
.
(
Ü
Ekeland
C
©
n
Ú
Û
¼
ê
½
n
,
·
‚
y
²
T
•
§
•
3
4
U
þ
C
Ò
)
(
¤
k
C
Ò
)
¥
ä
k
•
$
U
þ
ö
),
…
y
²
Ù
U
þ
0
u
Ä
U
þ
†
2
Ä
U
þ
ƒ
m
.
·
‚
ï
Ä
ò
r
?
Choquard
•
§
(
=
‘
k
š
Û
Ü
‘
Schr¨odinger
•
§
)
ï
Ä
,
•
þ
f
å
Æ
!
z
Æ
!
v
à
Ô
n
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z
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u
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A^
ê
Æ
?
Ð